Preface: I am currently in college in my first year of maths, so some more intricate explanations may not be available to me.
The situation I am modelling is that of an engineer repairing a machine, the more work he does, the less of a result it has overall, a classic diminishing returns problem. I would like to represent the machines efficiency (how repaired it is) as a percentage or a number from one to one hundred.
A logarithmic function $r=\ln(w+1)$ can be used to describe the diminishing returns relationship between work done ($w$, in hours) and result ($r$). My problem lies in the fact that result tends to infinity, where I would prefer it tends to 100: $\lim\limits_{w\to \infty} r=\ln(w+1)=100$
In my understanding of maths, to represent a number from 1 to n as a percentage, you would do $\frac{number}{n}*100$, but obviously I cannot simply divide by infinity. How could I modify this function so that it tends to 100 instead of infinity?
You could use something like $$ r(w) = 100 \left(1-\frac{a}{w-a} \right) $$ and tweak the value of the parameter $a$ to your liking, for example $a=100$.